§5 Currents after closing and opening the circuit
Consider a circuit consisting of a source current EMF ?, the resistance R, inductance L and the key K. The circuit current flows
t = 0 disable EMF, closing the circuit shorted (key in position 2). As
soon as the current begins to decrease self-induced emf arises that
prevents this decrease.
Current in the circuit satisfies Ohm's Law:
At t = 0
-time constant determines the rate of decrease (increase) in the circuit
therefore, τ - the time during which the current in the circuit is reduced by a factor e. The more or less L R (ie, the greater τ), the slower the current in the circuit decreases.
After connecting to the EMF until current reaches a steady state value
2. Current at closure.
, in the chain, except EMF E will act self-induced emf, which prevents the instantaneous current rise.
Then Ohm's law can be written as:
inhomogeneous linear differential equation.
is the total solution of the homogeneous differential equation
and any particular solution of the inhomogeneous equations.
The general solution homogeneous equations
Particular solution of the nonhomogeneous
The general solution of the nonhomogeneous
at t = 0 I = 0, hence , I0 = const,then
§6 Mutual induction.
two fixed circuit (1 and 2), located close enough to each other. If the
loop current flows one I1, then the magnetic flux generated by this
current is proportional to I1. We denote the part of the Ф21 thread that runs through circuit 2. Then
where L21 - proportionality factor .
If the current I1is changed, the circuit 2 is induced emf Ei2 2
Similarly, during the flow of current in the circuit 2 I2 magnetic flux pervades circuit - Ф12
If I2is changed, then
The phenomenon emergence the EMF in a circuit when changing current in the other is called mutual induction. Proportionality coefficients L12 and L21are called mutual inductance
L12 = L21.
L12 and L21are dependent
on the geometric shape, size, relative position of the contours and
the magnetic permeability of the medium surrounding contours
mutual inductance of two coils wound on a common toroid. Magnetic
induction field generated by the coil 1 to the number of turns N1,the magnetic permeability μ core
where is the length of the core in the midline
The magnetic flux through one turn of coil 2: :
The total magnetic flux
2. Transformers (converter) - a device to increase, decrease, electrical isolation, etc. voltage and current.
It is based on the phenomenon of mutual induction. Primary and secondary windings with N1 and N2turns,
respectively, mounted on a closed iron core. Change in the flux in
the coil 1 is connected to a variable emf is in the secondary
appearance EMF mutual induction, and in the primary - EMF of
self-induction. Primary current I1is determined by Ohm's law
R1 -the resistance of the primary winding ,
I1R1 small in comparison and in variable fields. Consequently,
EMF mutual induction in the secondary
Sign “ - “means that the phases and opposite .
- transformation ratio.
k > 1- step-up transformer ;
k < 1- step-down transformer.
Therefore, the currents in the windings are inversely proportional to the number of turns in these coils.
A transformer consists of one winding is called auto-transformer.